We investigate the short-time quantum critical dynamics in the imaginary-time relaxation processes of finite-size systems. Universal scaling behaviors exist in the imaginary-time evolution. In particular, the system undergoes a critical initial slip stage characterized by an exponent $\ensuremath{\theta}$, in which an initial power-law increase emerges in the imaginary-time correlation function when the initial state has a zero order parameter and a vanishing correlation length. Under different initial conditions, the quantum critical point and critical exponents can be determined from the universal scaling behaviors. We apply the method to the one- and two-dimensional transverse field Ising models using quantum Monte Carlo (QMC) simulations. In the one-dimensional case, we locate the quantum critical point at
(h/J)c=1.00003(8) in the thermodynamic limit, and we estimate the critical initial slip exponent $\ensuremath{\theta}=0.3734(2)$ and the static exponent $\ensuremath{\beta}/\ensuremath{\nu}=0.1251(2)$ by analyzing data on chains of length
L=32−−256 and 48--256, respectively. For the two-dimensional square-lattice system, the critical coupling ratio is given by
3.04451(7) in the thermodynamic limit, while the critical exponents are $\ensuremath{\theta}=0.209(4)$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.518(1)$ estimated by data on systems of size
L=24--64 and 32--64, respectively. Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts due to the essential differences between classical and quantum dynamics. The short-time critical dynamics and the imaginary-time relaxation QMC approach can be readily adapted to various models.
We propose a scaling theory for the universal imaginary-time quantum critical dynamics for both short and long times. We discover that there exists a universal critical initial slip related to a small initial order parameter
M0. In this stage, the order parameter
M increases with the imaginary time $\ensuremath{\tau}$ as $M\ensuremath{\propto}{M}_{0}{\ensuremath{\tau}}^{\ensuremath{\theta}}$ with a universal initial-slip exponent $\ensuremath{\theta}$. For the one-dimensional transverse-field Ising model, we estimate $\ensuremath{\theta}$ to be
0.373, which is markedly distinct from its classical counterpart. Apart from the local order parameter, we also show that the entanglement entropy exhibits universal behavior in the short-time region. As the critical exponents in the early stage and in equilibrium are identical, we apply the short-time dynamics method to determine quantum critical properties. The method is generally applicable in both the Landau-Ginzburg-Wilson paradigm and topological phase transitions.
While imaginary-time evolution has long served as a standard paradigm for ground-state preparation in numerical simulations and quantum devices, its intrinsic dynamical properties has been largely overlooked. Here, we investigate the short-imaginary-time critical dynamics in quantum systems with boundaries. A universal scaling theory is developed and verified in the two-dimensional quantum Ising model, uncovering rich dynamic critical behaviors dictated by boundary universality classes. For ordered initial states, the boundary order parameter
Ms decays with imaginary time
τ as
Ms∝τ−β1/νz, where
β1 denotes the boundary order parameter exponent, and
ν and
z correspond to the correlation length exponent and the dynamic exponent, respectively. For disordered initial states, the autocorrelation of the boundary order parameter is governed by a novel critical exponent
θ1, which is closely related to the critical initial slip behavior of
Ms characterized by the corresponding exponent
θ1′. In contrast to its positive bulk counterpart, the boundary initial-slip exponent
θ1′ is negative for the ordinary transition while remaining positive for the special transition. Although the static universality classes of
d-dimensional quantum phase transitions generally coincide with those of
(d+1)-dimensional classical phase transitions, we show that
θ1 does not follow this conventional quantum-classical mapping. We further discuss the implications of our results for more exotic forms of boundary criticality. Our findings provide new physical insights into boundary critical dynamics and offer a novel route for probing exotic boundary critical behaviors in quantum many-body systems.
We investigate the imaginary-time relaxation critical dynamics of the two-dimensional transverse-field Ising model using infinite projected entangled pair states (iPEPS) with the full-update strategy. Simulating directly in the thermodynamic limit, we explore the relaxation process near the critical point with two types of initial states: a fully polarized state and a product state with a small magnetization. For the fully polarized state, the magnetization shows a power law scaling
M∝τ−β/(νz) in the imaginary-time evolution, from which both the critical point and critical exponent can be determined with high accuracy. For the nearly paramagnetic state, the relaxation process exhibits a behavior of
M∝τθ with
θ=0.1958 being the critical initial-slip exponent, which is in good agreement with that obtained from the dynamic scaling of the self-correlation in quantum Monte Carlo method. These universal features emerge well before the system converges to the ground state, demonstrating the efficiency of imaginary-time evolution for probing quantum criticality. Our results demonstrate that iPEPS can serve as a robust and scalable method for studying dynamical critical phenomena in two-dimensional quantum many-body systems.
We explore the imaginary-time relaxation dynamics near quantum critical points with semi-ordered initial states. Different from the case with homogeneous ordered initial states, in which the order parameter M decays homogeneously as M ∝ τ −β/νz , here M depends on the location x, showing rich scaling behaviors. Similar to the classical relaxation dynamics with an initial domain wall in model A, which describes the purely dissipative dynamics, here as the imaginary time evolves, the domain wall expands into an interfacial region with growing size. In the interfacial region, the local order parameter decays as M ∝ τ −β/νz , with β 1 being an additional dynamic critical exponent. Far away from the interfacial region the local order parameter decays as M ∝ τ −β/νz in the short-time stage, then crosses over to the scaling behavior of M ∝ τ −β/νz when the location x is absorbed in the interfacial region. A full scaling form characterizing these scaling properties is developed. The quantum Ising model in both one and two dimensions are taken as examples to verify the scaling theory. In addition, we find that for the quantum Ising model the scaling function is an analytical function and β 1 is not an independent exponent.